4 ANALYTICAL MECHANICS Solving these we get tan a 34. DISCUSSION.—The last expression gives the value of « for a given value of M- When /z = 1, a. = 0, therefore in this raw the ladder will be in equilibrium at any angle between 0 and 4> with the1 ground. Kvidently this is true for any value of /* greater than unity. 3. Find the smallest force which, when applied at the center of a carriage wheel of radius a, will drag it over an obstacle, The forces acting on the wheel are: its weight W, the required force F, and the reaction R. Since the first two moot at the* center of the wheel, the direction of R must pass through the center also. Take I he eoorditwf e axes along and at right angles to R, as shown in Fig. 3«">, and let F make an angle 0 with the z-axis. Then the equations of equilibrium become IX m F COS $ - R + W COH « -- 0, SF mptinG— IF sin a = 0, Zffo's W • a sin a — F sin 0 «a =- 0. Prom either of the last two equations we get F* „ ?»i.....« w sin 0 Since W and a are fixed F can be changed only by changing 0. Therefore the minimum value of F is given by the maximum value of nia 0, i.e., 0 = -, which makes F^W&ina.